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Indian Journal of Pure and Applied Mathematics : Volume 48

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Indian Journal of Pure and Applied Mathematics : Volume 41

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Normed algebras of differentiable functions on compact plane sets

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Normed algebras of differentiable functions on compact plane sets

Content Provider	Springer Nature Link
Author	Dales, H. G. Feinstein, J. F.
Copyright Year	2010
Abstract	We investigate the completeness and completions of the normed algebras (D $^{(1)}$(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D $^{(1)}$(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which the completeness of (D $^{(1)}$(X), ‖ · ‖) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ.In an earlier paper of Bland and Feinstein, the notion of an $$ \mathcal{F} $$ -derivative of a function was introduced, where $$ \mathcal{F} $$ is a suitable set of rectifiable paths, and with it a new family of Banach algebras $$ D_\mathcal{F}^{\left( 1 \right)} \left( X \right) $$ corresponding to the normed algebras D $^{(1)}$(X). In the present paper, we obtain stronger results concerning the questions when D $^{(1)}$(X) and $$ D_\mathcal{F}^{\left( 1 \right)} \left( X \right) $$ are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever X is ‘ $$ \mathcal{F} $$ -regular’.An example of Bishop shows that the completion of (D $^{(1)}$(X), ‖ · ‖) need not be semisimple. We show that the completion of (D $^{(1)}$(X), ‖ · ‖) is semisimple whenever the union of all the rectifiable Jordan arcs in X is dense in X.We prove that the character space of D $^{(1)}$(X) is equal to X for all perfect, compact plane sets X, whether or not (D $^{(1)}$(X), ‖ · ‖) is complete. In particular, characters on the normed algebras (D $^{(1)}$(X), ‖ · ‖) are automatically continuous.
Starting Page	153
Ending Page	187
Page Count	35
File Format	PDF
ISSN	00195588
Journal	Indian Journal of Pure and Applied Mathematics
Volume Number	41
Issue Number	1
e-ISSN	09757465
Language	English
Publisher	Springer-Verlag
Publisher Date	2010-04-11
Publisher Place	India
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Normed algebra differentiable functions Banach function algebra completions pointwise regularity of compact plane sets Numerical Analysis Applications of Mathematics Mathematics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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